EXAMPLE 1.4.11

How many different 4-digit numbers can be formed using the following digits?

{0, 2, 3, 5, 8}

SOLUTION

In order to form a 4-digit number we must make must make 4 decisions:

i. Choose first digit: 4 options (the first digit could be 2, 3, 5, or 8).

ii. Choose second digit: 5 options (the second digit could be 0, 2, 3, 5, or 8).

iii. Choose third digit: 5 options (the third digit could be 0, 2, 3, 5, or 8).

iv. Choose fourth digit: 5 options (the fourth digit could be 0, 2, 3, 5, or 8).

According to the Fundamental Counting Principle the number of outcomes is

(4)(5)(5)(5) = 500. There are 500 possible 4-digit numbers.

How many different 4-digit numbers that are multiples of 5 can we form?

In order to form one of these numbers, again we need to make 4 decisions. However, in this case the last digit must be either 0 or 5 (in order for the number to be a multiple of 5).

i. Choose first digit: 4 options (the first digit could be 2, 3, 5, or 8).

ii. Choose second digit: 5 options (the second digit could be 0, 2, 3, 5, or 8).

iii. Choose third digit: 5 options (the third digit could be 0, 2, 3, 5, or 8).

iv. Choose fourth digit: 2 options (the fourth digit must be 0 or 5).

According to the Fundamental Counting Principle the number of outcomes is

(4)(5)(5)(2) = 200. There are 200 possible 4-digit multiples of 5.